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Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation
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  • Published: 18 September 2008

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

  • Lung-Chi Chen1 &
  • Akira Sakai2 

Probability Theory and Related Fields volume 145, pages 435–458 (2009)Cite this article

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  • 7 Citations

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Abstract

We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α > 0 converges to \({e^{-C|k|^{\alpha\wedge2}}}\) for some \({C\in(0,\infty)}\) above the upper-critical dimension \({{{dc \equiv 2(\alpha \wedge 2)}}}\). This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151–188, 2008). Moreover, we show that the constant C exhibits crossover at α = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Fu-Jen Catholic University, Hsinchuang, Taipei Hsien, 24205, Taiwan

    Lung-Chi Chen

  2. Creative Research Initiative “Sousei”, Hokkaido University, North 21, West 10, Kita-ku, Sapporo, 001-0021, Japan

    Akira Sakai

Authors
  1. Lung-Chi Chen
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  2. Akira Sakai
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Correspondence to Akira Sakai.

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Cite this article

Chen, LC., Sakai, A. Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation. Probab. Theory Relat. Fields 145, 435–458 (2009). https://doi.org/10.1007/s00440-008-0174-6

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  • Received: 28 April 2008

  • Revised: 09 July 2008

  • Published: 18 September 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0174-6

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Keywords

  • Long-range oriented percolation
  • Mean-field critical behavior
  • Limit theorem
  • Crossover phenomenon
  • Lace expansion
  • Fractional moments

Mathematics Subject Classification (2000)

  • 60K35
  • 82B27
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